60. Two Methods
c. Verify that your answers to parts (a) and (b) are consistent.

Verified step by step guidance

1. Review the answers you obtained in parts (a) and (b) carefully, noting the expressions or values you found for the quantity or function in question.
2. Identify the key results from both parts, such as derivatives, integrals, or function values, depending on what parts (a) and (b) asked you to find.
3. Set the expressions from parts (a) and (b) equal to each other or compare them directly to check for consistency. This might involve simplifying both expressions to a common form.
4. If the expressions look different, try algebraic manipulation such as factoring, expanding, or using trigonometric identities to see if they can be shown to be equivalent.
5. Conclude that the answers are consistent if you can demonstrate that both methods lead to the same result or expression, confirming the correctness of your solutions.

Key Concepts

Verification of Solutions

Verification involves checking that the solutions obtained from different methods satisfy the original problem or equation. This ensures consistency and correctness by substituting the solutions back into the initial conditions or equations.

Multiple Solution Methods

Using more than one method to solve a problem helps confirm the accuracy of results. Common methods in calculus include analytical techniques, graphical analysis, or numerical approximation, each providing a different perspective on the solution.

Consistency in Mathematical Results

Consistency means that different approaches yield the same or compatible results, reinforcing the validity of the solution. It is a fundamental principle in mathematics to cross-check answers to avoid errors and ensure reliability.